3 Answers
3

Even if you want to push, you won't actually be able to destroy a tower during the early game, and it can kill you quite fast. So while it seems intuitive to push when you are at advantage, it's a strong limit preventing you to push further, even if the lane goes good for you. This leads to the next point...

Positioning

When you push, the distance to your tower (and so, your safe place) is increasing, while it's decreasing for the enemy. As such, it will be complicated to actually damage them and even more to kill them. And at the same time, you are more exposed and have a greater distance to walk if things go wrong.

Minding the jungler

You might not have this situation in early levels, but the current metagame is such that you normally have one player in each team roaming in the jungle, and ganking lanes.

As such, by pushing your lane, you expose yourself to the enemy jungler while the enemy is protected from your own jungler, safe at their tower.

Optimizing your farm

In the current metagame (at least for western countries, it's a bit different when you see korean players, for example), there is a phase of heavy farming at the beginning of the game. To optimize this farm, you need to be safe, and have the minions accessible.

That's why pushing too hard, and killing the enemy tower too early can be a problem. In early game, the enemy tower helps you with farming, because it limits the distance where your creeps are going. Since there is no denying in LoL, it's actually the only way to have a sure limit for them.

Without the tower, your farm area is easily pushed farther in the enemy territory, which doubles the risks from the two previous points.

The only advantage I can think of pushing your lane is to force the enemy at their tower. Farming under the tower is complicated, as the tower will take a lot of the last hits on creeps. By pushing the enemy under their tower, you can deny them a fair amount of gold.

But it remains risky, and it will be enough for their jungler (or mid-laner) to come to your lane, to force you to retreat (unless you really have such advantage that you can take them 1 vs 2 or 2 vs 3, depending on the lane). So even if it looks like you are at an advantage, you might actually be offering them an easy kill or two. So careful with that, as the matchmaking level increases.

In most cases you will not want to push the lane since this makes you
open to ganks, however there are some situations where it does make
sense:

You notice the enemy team has gone for an objective (they are all
pushing another lane, doing dragon) and you will not be able to reach
the fight in time. You would be better off pushing your lane (so that
the enemy champion that would have been in the lane loses experience,
and if you push hard enough, you might even be able to get the tower).

When you have your summoners up and perhaps a built-in escape. If you
push the lane on your enemy, they are more likely to miss minion kills
at tower (it is harder to farm at tower effectively). This makes you
susceptible to ganks, however if you feel comfortable that even if you
were ganked you would be able to evade, then it should be okay.

When
the enemy team is going to be very ‘late’ to lane. This is common at
level one. Sometimes the enemy champions from top or bottom lane will
stay to protect their jungler, by doing so they will also be late to
the laning phase. If you are able to push the lane, they could miss
one or two waves of minions. This would give you a significant
advantage in lane and well worth the push (even if you are more open
to ganks).

When not to Push

You know that the enemy jungler has finished their jungle route (After
3:50/4:00 on the game clock) and is looking to gank the lane

When you
don’t know where the jungler is (you do not have map awareness)

You
don’t have your summoner spells available (they are on cooldown)

When
your enemy jungler wants to gank the lane and you want the lane to
push to your tower.

When to Keep Lane Even

This means that you are trying to keep the creep line about even with
your opponent, the minion line will not push to either tower, you will
simply keep equidistant between the two towers.

When you think that the enemy Jungler may be on your side of the
jungle (the mid lane splits those sides (top and bottom)).

When your
allied Jungler has no plan to gank the lane anytime soon.

When your
opponent is hitting the creeps more than once (they are pushing the
lane on you) You will want to match the amount of damage they are
doing to the creep line. This might mean you would use abilities to
clear minion waves if your opponent is doing the same. Keep the same
pace so that the minion lane stays in the middle.

Make your opponent loose as much gold as possible (kill him/ prevent him from farming/surving)

Now here is the question : Does killing the turret will help to reach these objectives ?
(Will it raise the chance of dying/ killing ? Will it help you to farm or your opponent to farm?)

Yes it could if you can "Freeze" your lane near your tower then the ennemy will have to go far from his tower to take the cs and xp.

But if it's him who "Freeze" his lane near his tower you will have to take risks to get there.

Still it depends on your ability to escape : champions like Singed or Jayce don't even care about ganks but some like Ashe or Kog'maw would be very sensitive to overextend

As an AP MID caster I would say that pushing your tower is not a bad idea because it allows you to roam and if you over extend there is 2 ways to run from a gank.

but in the sides lanes it's more problematic looking at the distance to your safety.

What you have to consider too is that some players likes to hug their tower, and considering the damages they do it's better to fight under the outer turret than the middle turret. and the distance beetween the 1st and the 2nd is longer than beetween 2nd and 3rd.

IMO most of the time the best thing to do is to crush your minions on the enemy tower instead of killing the tower and use the free time to get full life/mana or to do some jungle like wraith, wolves, golems and the most important help your teamates (Mostly as an AP MID ;-) )!